CHAPTER- 6 Some Astronomical Principles : an Indian Perspective

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CHAPTER- 6

Some Astronomical Principles : An Indian Perspective

Two types of knowledge: apariividyli (inferior knowledge) and parlividyli (superior or spiritual knowledge) have been accepted in Indian tradition. An act of worship done

with a specific worldly desire was .consid~red an inferior form of worship and was popular with the kings. Aparlividya enables man to attain material progress, enrichment and fulfilment of life and parlividyli ensures attainment of self-realization or salvation in life (Chand. Up. 7.1.7; MuYJc!a. Up. 1.2.4-5). The Vedic people, in general, made interesting synthesis by adopting nitya (perpetual or daily) and klimya (optional for wisl¥ fulfilment) sacrifices or offerings. The first was supposed to .bring happiness to the family and second was for material progress. The perpetual daily sacred fires and the optional fires were placed on altars of various shapes. As to the reasons, which might have induced the ancient Indians to devise all these strange shapes, the J?,g-Veda (1.15.12) says, 'He who desires heaven, may construct falcon shaped altar, for falcon is the best flyer among the birds'. These may appear to be superstitious fancies but led to important contributions in geometry and mathematics because of their conviction in · social value systems.

To find the right time for religious, agricultural, new.year and other social festivals gave the motivation for recording of recurrence of repeated events from seasons, stars, movement of planets, moon etc. This helped to develop many a framework for mapping of movements of heavenly bodies with reference ·to East, West, 'North and South points, nak:jatras, calendar, yuga, inahliyuga and movement of planets for mean and true positions of planets. Various mathematical 1and trigonometrical tables were also formulated for better and better results. The Vedlinga Jyoti~a 1 mentions as under:

102 Having saluted Time with bent head, as also Goddess Saraswati, I shall explain the lore of Time, as enunciated by sage Lagadha. As the crests on the head of the peacocks, the

jewels on the serpents, so is the (jyoti~a ga1]itam) held at the head of alllores among all vedliY!ga slistras.

G~ita is a variant reading for jyoti~a meaning computation, which is the essence of this science.

Another tradition, which has enriched specialized activities in mathematics and

astronomy is the guru-si~ya-paramparli (teacher-student tradition). Different recensions of Vedic schools, sulbakaras, jyoti~karas (Varahamihira names tWenty scholar before him), Kusumpura school, schools of Ujjain and Asmakadesa, and Jain and Buddhist schools are also well-known in this coimection.

Beside these, there were commercial and ot9er problems, which were tackled for­ lokavyavahlirlirtha, used for 'common people. The restriction and emphasis were also assured on social use and value systems, which helped people to take up different activities for commerce, education and other areas. These helped undoubtedly to concretise knowledge resulting in original contributions to mathematics and astronomy.

Construction of altars and nature of knowledge :

The ritual connection of Indian geometry, as elaborated by Thibaut and Burk, has been intensively discussed by Datta, Seidenberg, Sen and Bag, ~d others.2 The ceremonies were performed on the top of altars built either in the sacrificer's house or on a nearby plot of ground. The altar is a specified raised area, generally made of bricks for keeping the fire. The fire altars were of two types. The perpetual fires (nitya agni) were

constructed on a smaller area of one Sq. puru~a and optional fires (klimya agni) were

constructed on a bigger area of 7Y2 sq. puru~as or more, each having minimum of five ·layers of bricks. The perpetual fires had twenty-one bricks and optional fires had two hundred bricks in each layer in the first construction with other restrictions. For optional

103 fire altars, the•. whole family• of the organizer had1 to reside by the construction site of optional fire altar, for which another class of structure known as mahiivedf and other related vedfs were made. However, a summary of these types of altars with ground I shapes are grouped below:

(a) Perpetual fire altars (area coverage: 1 sq. puru~a): iihavanfya (square),

giirhapatya (circle or square), da~ii'Jiigni (semicircle).

(b) Optional fire altars (area coverage: 7-~ square puru~a): Caturasrasyenacit (hawk bird with square body, sq uare wmgs, square tail), kankacit and alajacit (bird, with curbed wings and tail), prauga (triangle), ubhayata prauga (rhombus), rathacakracit (circle), drafJacit (trough), smasiinacit, (isosceles trapezium), kurmacit (tortoise) etc.

(c) Vedis: mahiivedf or saumikf vedf (isosceles trapezium), sautramani vedf'

(isosceles trapezium, and also one-third of the mahiivecff); pait~kr vedf (isosceles trapezium or a square, area one-third of sautramani vedl), pragvan:zsa . (rectangle).

One can guess the natUre of knowledge, which could originate from such altar constructions. However, the Sulba-Sutras of Baudhayana, Apastamba and other schools have summarized this knowledge as available from the SaiT).hitas and Brahmaqas. Both Baudhayana and Apastamba belonged to different schools but followed a similar pattern, which also suggests that these schools inherited the knowledge from older . ~ , schools. While giving details, the Sulba-Sutras use the word vijfziiyate (known as per traditions), vedervijfziiyate (known as per Vedic tra~ition) etc. very often. A summary of this knowledge \\fill be of great interest.

Baudhayana gives various units of linear measurements viz., 1 pradeia = 12 angulas; 1 pada = 15 ang; i~a = 188 ang; 1 a~a = 104 ang; 1yuga = 86 ang; 1 janu = 32 ang; 1 samyii = 36 ang; 1 biihu = 36 ang; 1 prakarma = 2 padas; 1 aratni = 2 pradesas; 1 puru.$a = 5 aratnis; 1 vyiiyama = 4 aratnis; 1 ang = 34 tilas = % inch (approx.).

104 Knowledge of rational numbers like 1, 2, 3 .. ~ 10, 11 ... 100 ... 1000, Yz, ~. lf4, Ys, 1116, 3/2, 5/12, 7Yz, 8Yz, 9Yz etc. were used in decimal word notations and their fundamental operations like addition, subtraction, multiplication and division· were carried without -:-·· ·any mistake.

I Baudhayana had knowledge of square, rectangle, triangle, circle, isosceles, trapezium and various other diagra~s and transformation of one figure into another and vice­ versa. Methods of construction of square by .adding two squares or subtracting two . i squares were known. The areas of these figures were also calculated correctly. That the length, breadth and diagonal of a right triangle maintain a unique relationship, a2 + b2 = c2 (where a = length, b = breadth and c = hypotenuse) or in other words ( 1, 1, 2) (2, 1, 3) formed important triplets thereby forming an important basis for number line, and were used for construction of bricks and geometricai figures. For easy verification, Sulbakaras suggested triplets expressed in rational and irrational numbers like (3, 4, 5),' (12, 5, 13), (15, 8, 17), (7, 24~ 25), (12, 35, 37), (15, 36, 39), (1, 3, -../10), (2, 6, -../40), (1, -../10 -../11), (188, 78'lS, 203o/J), (6, 2Yz, 6Yz), (10, 41/6, 105/6) and so on. A general statement on 'Theorem of Square on Diagonal' was also enunciated thus: 'The areas (of the squares) produced separately by the length and the breadth of a rectangle together equals the area (of the square) produced by the saine diagonal'. This has been wrongly . referred to Pythagoras who was assoCiated with the triplet (3, 4, 5) and the theorem is known as the Pythagorian theorem. Indian knowledge is based on rational and irrational arithmetical facts and geometrical knowledge of transformation of area from one type to the other and its importance was perhaps correctly and perfectly realised. How the Babylonians, Egyptians, Chinese and the Greeks came upon the knowledge of ·triplets but not the general statement is equally important for an interesting study. 3

The Sulba-Sutra tradition vanished. Only a limited commentary from a late period is '. available. Whether the tradition has been lost or the elements· have been absorbed in temple architecture is still to be investigated and may be the part of other studies. . . I

105 Decimal scale, decimal place-value, numerical symbols and zero :

'Our numerals. and the use of zero', observes ,Sarton4 (1955), 'were invented by the

Hindus and transmitted to us by th~ Arabs (hence the name Arabic numerals which we gave them)'. The study of Sachs, Neugebauer on Babylonian tablets, Kaya and Carra de Vaux on Greek sciences, Needham on Chinese sciences and study of Mayan culture have many interesting issues. The study of scholars likes Smith and Karpinski, Datta, Bag and Mukherjee5 have analysed Indian contributions, but still there is a need for a comprehensive volume. However, the salient points may be of interest: The Indians had three-tier system of word-numerals starting from the SamhiUis as follows:

·(a) Eka (1), dvi (2), tt; (3), catur (4),pafica (5), ~a( (6), sapta(7), a~{a (8),and nava (9).

(b) Dasa (10), vimsati (2 x 10), trimsat (3 x 10), catviirimsat (4 x 10), paficiisat (5 x

10), ~a~thf (6 x 10), saptati (7 x 10), asiti (8 x 10) and navati (9 x 10).

2 4 (c) Eka (1), dasa (10), sata (10 ), sahasra (10\ ciyuta (10 ), niyuta (10\ prayuta 6 8 9 10 11 ( 10 ), arbuda (1 0\ nyarbuda ( 10 ), samudra ( 10 ), Madhya (1 0 ), anta (1 0 ) 12 andpariirdha (10 ).

The names and their order have been agreed upon by almost all the authorities for (a) and (b), whereas there is a variation in (c) where mostly one or two terms have been I added later. The numbers below 100 were expressed with the help of (a) and (b)

sometimes following additive or substractive principles e.g., trayodasa (3 + 10 = 13),

unavimsati (20- 1 = 19), while for numbers above hundred, groups (a), (b) and (c)

were used. For example, sapta satiini vimsati = (720), .ra.rthim sahasrii navatim nava

= (60, 099).

One feature of the application of the scale is that it has been used in higher to lower order (sahasra, sata, dasa and lastly the eka). Real problem started when the numerical

symbols began to appear. The a~(akarYJi or a~(c;tmt;dam fairly indicates that Vedic people identified eight marks but whether they identified other symbols is not known. The

106 Mahabharata (III. 132-134) narrates a .story in which it says that 'The signs of 2 calculation are always only nine in number • The a~tadhayf of PiilJini (4SO BC)·used the word lopa, and Patafijali the word su~ya in connection to metrical calculations. When -

Brahrn.I and Kharo~thi numerals/alphabets appeared on the scene, there were lot of confusions creating more problems for ordinary business people and the mathematicians and astronomers as to how to use the numeridl symbols and adjust with the existing decimal system. The early inscriptions show the number system was additive and did not use decimal scale. Moreover numerical symbols were many in the beginning and it was difficult to decipher the correct meaning.

First attempt of a synthesis of the Vedic decimal system with the prevalent situation was possibly made by the Jains. The Anuyogadvarsiitra (100 BC) has described the numerals as anka as we find in Bengali to mean mathematics and describes decimal scale as decimal places (ga1Janasthana) and their numeral vocabulary was analogous to, that of the Brahmanic literature. They have enlarged these places to 29 and beyond, and we find more clear sta~ement ·in mathematics cum astronomical texts from Aryabhq(ta .onwards in expressions like sthanasthanam dasagu1Jam syat (from one place to next it should be ten times) and dasagu1Joltariih sam}Piah (the next one is ten times the previous one). This indicates that the scale was merged with the places, and the system became very simple. For example, the Vedic numbers: sapta satani vimsati and ~a~(him sahasra navatim nava reduces to: sahasra ( 10\ dasa (10), eka (1) Places

7 2 0 = 720 .60 0 9 9 = 60,099

The order of the Vedic scale was from higher to lower (sahasra, sata, da~a and eka). 1 But later, tlie order of the scale was changed frhm left to right i.e., eka, da~a, sata etc. This is obvious when we :think that in the Vedic system all words were spoken, and in the latter system the scale obviously followed the written style (that is from left to right)

107 and the place values were from eka to higher order. Moreover, the symbols were not standardized and interpreted differently in different regions. To avoid this problem, experts coined synonymous words and used them as symbols in decimal place-value in, lower to higher order and the actual number was obtained by reversing the number. For example: sunya (0), dvi (2),panca (5),yama (2) was actually 2520.

The Kashmiri Atharva-Veda also uses similar symbols.

Association of decimal scale with place-value was so popular in Indian tradition that it was not even commented upon. The popularity went to deep that even Sankaracarya (c. AD 800), the great social reformer, pointed. out that the same numerical sign if placed in unit, tenth, and hundredth places becomes 1, 10, 100. The men in business or in elementary schools (pi'ithsalii) used wooden board (pii,tf), hence the name pa(fgatJita, for quick calculation, in which dust was spread and finger or hard materials were used for , calculation. The system also movecl-to Java, Malaya, and other East. Indian colonies along with the busin_ess people, which is evidenced from some available inscriptions.

The use of decimal place-value in lower. order ~ith word-numerals and higher to lower order with numerical symbols was in practice. For calculation on a pli(f, numerical symbols were used, but for writing or copying a manuscript, final results were written in word numerals to avoid confusion in decoding a symbol and also to keep rhythm in verses in which it was written. The zero in many places of Bakshalf. Ms (AD 400) has been used as a round symbol (sunya, 0). It also came out as dot(·), may be, that thick tip of pen used for circle became dot in the process. This is distinctly visible in Bakshalf Ms and the Kashmiri Atharva-Veda. Alberuni (c. 1020 AD) has incidentally referred to two systems of notation of numbers, namely alphabetic (abjd) system (Huruf al-jummal or HiSab al-jummal) and the Indian numerals (al-Arqam al-Hind). He has recorded Indian numerals of nine symbols, and zero as dot in the Kitab al-Tajhim (the book of instruction in the art of astrology). He also referred to circular symbol (0) of the Indian. Al-Khwiirizmif (825 AD), another Central Asian scholar, writes about Indian numerals6 thus, 'The beginning of the order is on the right side of this writer, and this will be the

108 first of them consisting of unity. If instead of unity they wrote X, it stood in the second digit and their figure was that of unity, they needed a figure often similar to the figure of unity so that it became known that this was X, and they put before it one digit and wrote it in a small cicle "0" so that it would indicate that the place of unity is vacant.'

The Indian name sunya was taken over by the Arabs as a~-~ifr. This was subsequently changed to zephirum (1202, Fibonacci), tziphra (1340, Planudes) and Zenero, zepiro (sixteenth century, Italy).

Astronomical Features:

The fire sacrifices. including const!llction of fire altars appearing in the Salflhittis, • Brtihmal}as and the Srauta-Sutras clearly indicate that agnihotrs were concerned mainly · with directions for laying the sacrificial fires, new and full moon, the season and accurate calculation of the times of the year· etc. The fixation ·of east-west and north­ south lines was considered very important for the construction of altars. Ktitytiyana says' that eastern and western shadow-points of a central pole in a circle on equinoctial day fix the east-west line· and the line perpendicular to that gives the north-south line. The Satapatha-Brtihmal')a ~eports the K~ttikti never deviated from the east. R.N. Apte, the well-known Vedic scholar feels that east point was verified by the rising point of

K~ttikti in the Vedic period. The Vedic people took Sun as the sole light-giver of the universe, the cause of the seasons, winds, controller and the lord of the world. The

Moon was described as suryarasmi, one whieh shines by the 0 .sun's light. Different phases of the moon viz., rtikti (full-moon day), anumati (preceding full-moon day), kuhu (next full-moon day), sinivtili (preceding; new-moon day) etc. was known. The

Taittirzya-Briihmal')a give8. a full list of name of 15 days of the light half (purva pa~a) and also of dark half (aparapa~a). The day was called vtisava or aha and reckoned . . . . . from sunrise to sunset. The day was further divi~.ed into different parts. The period from one moonrise to the next or from one moonset to the next was known a·s tithi, which is somewhat different from the present concept of tithi of fixed time. That the phenomenon of new and full-moon is related ~o moon's elongation from the sun was

109 also correctly guessed. The invisibility of the Moon on the new-moon day is explained by its being swallowed by the Sun and its appearance by its being released by the Sun.

Nak~atras, Months, Names ofSeasons and New Year:

The na~atras or the group of stars near which the moon could be seen are the conventional division or marker of the ecliptic, (the path) which is followed by Moon, Sun and the Planets. The Moon returns to the same position in more than 27 days but less than 28 days. There is mention of 28 na~atras early sar1Jhitas, Atharva-Veda and others, but the numbers were reduced to 27 from the time of Vedanga Jyoti~a. The names ofthese na~atras with their·iJ.residing deities are enumerated in the Yajur-Veda beginning with Kr,ttika. The lunar (or synodic month) was measured from full-moon to full-moon or from new-moon to new-moon (TS. 7.5.6). The Taitt, saf1Jhita (5.6.7) refers I to 12 or 13 lunar months of a year and calls the 13th (intercalary) month by the names ' San:zsarpa and Ar1Jhaspati (TS. 1.4.14). The six t:tus in solar year with names of 12 tropical months are in Taitt. Sam, (4.4.11.1) and Vajasaneyf sam (13.14). Studies have already been made on Rohil}i, Kr,ttika, BharaYJf and Asvinf legends by Tilak, Shamasastry, Sengupta, Dikshit, Kupannaswamy Sastry which indicate that the reference point on the ecliptic .on equinoxial day (when day and -night were equal) shifted to asterism RohiYJi, Kr,ttikli (Sa.Il1hita period), BharaY}f (Vedanga Jyoti~a period), asvinf (Surya Siddhtinta period) in course of time. A seal from the Mohenjo-daro (M. 2430) corroborates the Kr,ttikli legend7 which supports that the new year started from Agrahayanf after the full moon at Kt:ttika. By the time of Kausitakf-BrahmalJa (19.8) the new year started with the winter solstice (shortest day) and it was on the new moon day of Magha. This indicates that the pumimanta system (from the termination of full moon) changed to amanta system (from the termination of new moon) with time and the effort went on to synchronize lunar month with the beginning of the tropical year and seasons.

110 Nak11atras Lunar Solar Months & Seasons Taitt-Sart~ 4.4.1 0 Months Taitt-sart~ 4.4.11 Athar. Ved. 19.10 Vaj. Sart~ 13.14

1. K~;ttikii Kartika Urja Sarada (Aicyon, 360 long) (Autumn) 2. Rohini (Aldebaran, 460) 3. Mrg!lSirsa AgrahayanT Saha Haimanta (A. Orionis, 600) (Dewy) 4. Ardra (Betelguese, 650) 5. Punarvasu (Pollux, 900) 6. Pu11ya Pau1)a Sahasya (Cancrii, 1050) 7. Asle11a (Hydras, I 090) 8. Maghas . Magha Tapa Sis ira (Regulas, 1260) (Winter) 9. (Piirva) Phalguni Phalguni Tap~sya (o Leonis, 1380) 10. (Uttara) Phalguni (Denebola, 1480) 11. Hasta (o Corvi, 1700) 12. Citra Caitra Madhu Vasanta (Spica, 1800) Spring 13. Svati (Arcturus, 181 0) 14. Visakhe Vaisakha Madhava (Centauri, 2160) 15. Anuradhas (Scorpi, 2190) 16. Jye~tha Jai~tha Sukra Grr~ina (Antares, 2260) (Summer) 17. Miila (A. Scorpi, 241 0) 18. (Piirva) A~adhas A~acja Suci Sagittarii, 251 0) 19. (Uttara) A~adh!Ui Sagittarii, 2590) 20. Abhijit (Vega, 2620) 21. Sravana Sravana Nabha · Var~a (Altair, 2780) (Rainy) 22. Sravi11\hi'is (13 Delphini, 2930) 23. Satabhisa (A. Aquarii, 3180) 24. (Piirva) Pro~thapadas Bhadrapada Nabhasya (Markob, 3300) 25. (Uttara) Pro~thapadas (v Pegasi, 3460) 26. Revatr (f.h Piscium, 3560) 27. Asvaujau Asvina 1sa Sarada (13 Arietis, 100) (Autumn) 28. Bharanis (41Arietis, 250)

111 Luni-solar adjustments and time units:

The natural means of measuring a year originated from the experience of periodic recurrence of climatic seasons. Likewise, the natural means of measuring a day was the period between two full moons. The return of the Sun to the same position with respect to the fixed star might have appeared to be much more reliable than the slow seasonal . . . variation of the length of day light. There appears to be a constant attempt at adjusting the lunar month with the season .. The Taittirfya-Sa11]hitii (7.2.6) mentions how eleven I days ceremony (ekiidasaratra) were performed after lunar year of 354 days to make up with the seasons (r,tus) i.e., with the sidereal year of 365.25 days. The idea of intercalating a month at regular intervals of time or of adding of five or six days in one . month or more months was thus developed. Naturally, three units of time measurement viz., the solar day, the lunar month and the solar year are involved. Consequently the luni-solar adjustments depended on the problem of finding the integers x, y, z which, satisfy the relation, x years· = y months = z days. The f?.g- Veda gives two years (samvatsara, parivqtsara), Talttirfya-Briihmal') (1.4.1 0) mentions . four year~ (samvatsara, parivatsara, idavatsara and linuvatsara). Shamsastry also believed in a four year cycle, first three years of 360 days and fourth year= 14614 = 36514 days. The

VedangaJyoti~a gives (both ~gvedic recension- arcajyoti~a, 36 verses and Yajurvedic recension - yliju~a jyoti~a, 43 verses) a five-year cycle, the number of sunrising, moon rising, tithi, na~atra, the new-moon, full-moon, account for solstices, seasons, Sun's northward and southward journey, increase and decrease of day lengths during these journeys, rules for determining the beginning of the season, etc. It prescribes five solar years = 62 sidereal months (moon's revolution) = 67 lunar months (synodic) months. In this cycle offi_ve years, solar days= 5xl2x30 = 1800, civil days (solar rising_= 1830, sidereal days(solar rising plus solar cycle, earth's rotation)= 1835. A five year was conceived as yuga in Vediinga Jyoti~·a. The conception of caturyuga, and kalpa were later developments. The Vediinga Jyoiio¥a (Y-VJ, 24) also gives time units (measured by water clock), 1 iic/ikii =50 palas, 1 kuc/ava = l/I6 iic/ikii = 3 Ys palas, I niic/ikii = 200 !Jalas- ;3116lic/aka.<£.·= 4 iic/akas-3116 iic/akas = 6 1/16 iic/akas = 61 kucfavas, 1 niic/ika = I 0 1/20 kaliis, 60 niic/ikiis = muhiirtas = 1 day. 8

112 Astrological tradition :

The Atharva Jyoti,ra (in 14 chapters, 162 verses) attached to Atharva Veda tradition deals with the muhurta branch of astrology. Atharva Veda says that it was taught by Pitamaha to Kasyapa. The muhurta as a time unit comes first in the Satapatha Brtihmaf)a (12.3.2.5) which gives 1 divasa = 30 muhurtas, 1 muhurtas = 15 k,ripra, 1 k,rpra = 15 itarhis, 1 itarhi = IS idtinis, I idani = IS prtiY)as. The Atharva Jyoti~a gives the unit as ahoratra (whole day), muhurta, tru(i, kala, lava, former being 30 times the next instead of IS times. The last unit is nimesa ( 1 lava = 12 nimesa). R~ference to 7 planets, sun, moon and five planet~ are found. The Atharva-Veda (19.9) writes, 'May the Gandramasa and Aditya graha, along with Rahu prove auspicious to you'. The Atharva Jyoti,ra also gives names of seven week days with names of planets and describing planets as 'Lord of the days'. The names are aditya (sun), soma (moon), bhauma (mars), budha (mercury), b~:haspati (Jupiter), bhtirga~a (Venus) and sani, (Saturn). In other places it has also used names qf planets as being applicable to planets . , I like Surya (Sun), Lohittinga (Moon), Somasuta (Mercury), Devaguru (Jupiter), Bh~:gu, Sukra (Venus), Muryasuta -(Saturn). This also contains the seeds of prediction of Jataka astrology expressing fears, woes and horrors when certain nak,ratras are accompanied i by planets, meteors (ulka) etc. The Asvalayana Sutra contains instructions like fields should be ploughed on the uttara pro~(hapada', (G~hya-Sutra, 2.1 0.3), · 'thread ceremony should be performed on auspicious nak,ratras' (Gt:hya-Sutra, 1.4.1) etc. The words, muhurta and k,rana also appear in the 'nirukta'. The Ytijfiavalka Smt:ti gives the names of nine planets (seven planets, Rahu and Ketu (Acaradhyaya) and twelve parts of the ecliptic (rasis) in order to find proper times (Surya-san;karama) for performing the sradha. The Yavana Jtitaka of Sphujidhvaja9 (AD 269) is considered as a major work carrying Greek influence. But it has also used (last chapter) time units viz. palas, kucjavas, liptas, naqika, kala, muhurta, (or k.rafJa) same as Vedtinga Jyoti.ra, with the exception that it has used 1 nacfika = 10 kalas instead of 10 1/20 ktilas used by Vedanga Jyoti.ra, possibly to avoid fraction. It has conceived a cycle of 165 years instead of 5 years and has given the number of solar months, solar days, civil days, intercalary

113 . months, tithes, omitted tithes, sider~~l.months and other elements for this cycle. The.· · .. Bt:J;ajjlitaka (or Brhat Jlitaka) is -the next most important work on astrology by Varahamihira (c. AD 505) who also:· compared 'Brhat Jlitaka (horoscopic work) and three smaller works on marriage and ylitra (journey) on astrology, besides collection of five important works on astronomy (Paficas{ddhantikii). Varahamihira referred to Parasara _(twice, MlifJqavya and licaryas like Satya, Maya, Yarana, Manithya,

· Jivasarma, Vi~f}ugupta (identified as Cfu}akya Vi~fJugupta, the minister of Candragupta by the commentator). Bhaf(otpala (c. AD 966), the commentator on the_ Brhat Jataka ·has given astrological quotations fro.m- the works of Glirgf, Badarliyana, Yajfiavalkya and Maqqyavya. The Brhai Jlitaka ~ontains 36 Greek words namely, the 12 signs - kriya, taburi, jituma, kulir, leya, pathena, juka, kaurpya, tauksika, akokera, Hridroga, Ittham and twenty four other words including the words, liptli, horli_; dresklina, Kendra, . . ' . · kofJa etc; On the basis of these terms, Weber and Kern believe that there was Greek influence on Indian astrological works. Large· number of late astrological works on'

jatakas (Katacakra-Jataka, Gauri-Jataka, Minarajd"'Jataka, Jatakasara of N~hari, Saravaii-Jataka Paddhati etc.) dealing with misery and woe based on ascendants and descendents of planets, horlis (by Vasif!tha, Garga; Parasara, Varaha, Lalla, Narada etc.) ·. dealing with whether an event will happen at all, if so, when and how on the basis of

horoscope cast, muhurtas (tithes, na~atras prohibited for auspicious ceremony, taboos etc.), plisaka vidyli (questions answered according to the casting of dice), tlijik (forecasting as per ascendance and descenderice.ofplanets) etc. were in vogue.

Yuga, Kalpa and Mahayuga:

The Vedaf}ga Jyoti~a had a cycle of five years in a yuga. The Yavana Jlitaka had 165 years cycle. The PafJini, though refers to yuga· and koli, it is lvi!"Jnusmt:ti which gives

four yugas(K~ta, Trela, Dvapara and Kali) = 12000 divine years= 4320000 ordinary human years, since 1 Divine year= 360 ordinary years. The Mahabharata also had the same yuga and time units as that of Manusmt:ti. The Siddhantic astronomy, unlike Greek astronomy, has established on epoch when all planets were in zero longitude. Aryabhatta I considered on epoch when the Slfn, the Moon, Mars, Mercury, Jupiter,

114 Venus and Saturn were last in zero longitude at sunrise in Lanka (a hypothetical place at . the intersection ofthe equator and the meridian ofUjjain) on Feb. 18, Friday, 3102 BC. The period of one such epoch to the next, according to Aryabhatta I is 1,080,000 years. When the Moon's apogee and tlie Moon's ascending node are included in the list of the. planets, the above mentioned period (mahliyuga) be.comes 4,320,000 years, which is defined as the duration of yuga. The yuga is a period of time which begins and ends when the Sun, Moon, Mars, Mercury, Jupiter, Venus, Saturn, the Moon's apogee and the moon's ascending node are in zero longitude. It consists of 4 periods, Kr:ta, tretli, dylipara and kali, the current quarter yuga is the current kaliyuga which is assumed to have begun at sunrise at Lanka on :Friday, 18 Feb., 31.02 BC .. A ?Jigger period than the yuga is called kalpa. According to AryabhaUa I, kalpa consists of 1008 yugas, and 459% yugas had elapsed at the beginning of c~rrent kaliyuga since the beginning of· current ka/p{!. The main -difference in Aryabhatta, Surya-Siddhiinta and Brahmana Siddhanta School is that the length ofthe year and motion of planets in a kalpa and, mahliyuga is different and other elements are same. There is also difference of opinion as to the starting of kali era on 18 Feb., Friday, 3102 BC, which needs more careful scrutiny. The Hindu astronomical works called Siddhlinta adopt the terms of creation as the epoch of calculation whereas those called kaf:iyuga as the epoch of calculation.

Old Siddhantic Tradition :

The Jains did positive contributions to mathematics. A few works like Surya Prajfinapti, Candra Prajfiapti, Jamboo Divfpa Prajfiapati, Sihlinlinga Siitra, Bhagavatf Sutra, Anuyogadvlira Sutra are available to us. It deals with problems dealing with circle, chord, circumference, p (=10), diameter, arc, segment, big numbers, infinity, laws of indices, symbols, operations etc. Varahamihira was born in AD 505 in the . village of Kapithaka (Farrukabad district of Uttar Pradesh) and. moved to Ujjain. His forefathers migrated to India from Maga country in Persia and settled in the Kapithaka. He quoted Aryabhatta I several times and compiled Paficasiddhlintikli i.e., five

siddhanta works, namely, aulisa, romaka, vasi~(ha, saura and pclitlimaha besides astrological works. Colebrooke (1807-17), Whitney and Burgess (1860), Kern (1865),

115 · . Thibaut (1890) and a few other Europeap schblars passed judgement on the relative importance and origin of Indian· astronomy. Thibaut in his introduction to the Paficasiddhlintikli observes that the Paitlimaha Siddhiinta (c. AD 80) is the oldest and carries pre-scientific stage of astronomical knowledge. The Vasi,stha Siddhiinta, written prior to AD 269 is more advanced. The Romaka and Paulisa have Greek influence. The

Saura Siddhiinta only contains new featur~s. During the early centuries of the Christian era the Indians were .in touch with the Greeks, Romans and other scholars and those of Babylonian and Greek knowledge may have beeh available to them.

The scholars. like Dikshit, · Sengup~a, Ganguly, Kuppani:laswamy and Shukla11 testified that the refinements introduced by Ptolemy (AD 150) and even Hipparchus (ISO BC) · remained unknown to India. Whatever Greek influences are there, they are all of pre­ Ptolemaic period and possibly of pre-Hypparchus time. The extent and nature. of contact were through conferences or direct borrowal through translation of texts is still to be.

investigated.· Neugebaur has· shown that the Vasi~tha and Paulisa were inspired by Babylonian linear astronomy.

The Paiicasiddiintikli (five siddhiintas) were known in India from first century AD to fifth century AD. By this time, the Indians had already acquired the knowledge of zero and decimal place-value, eight fundamental operations of arithmetic addition,

subtraction, multiplication, division etc., rule of three,: inverse n."-le of three~ knowledge of combinations of six savours (a, b, c, d, e, f), 2 at a time, c (6,4)- ab, ac, ad, ae, af, be, ' bd, be bf, cd, ce, cf, de, df, ef- fifteen in all), 31 at a time c (6,3), 4 at a time c (6,4) was known. Likewise, the ·knowledge of binomial. expansion for calculating the shortcomings in metrical rhythm of music based on long (a) and short (b) sounds were 2 2 3 known. Or in other words binomial expansion like (a +, bi = l.a + 2a. b + 1. b ; (a + b ) 3 2 2 3 4 4 3 2 2 3 4 = l.a + 3.a b + 3.ab + I.b , (a+ b) = l.a + 4.a b + 6 a b + 4ab + l.b and achieved various other mathematical results. These undoubtedly brought great change in the

Indian scenario in the field of mathematics~ and astronomy. The development of algebraic and trigonometric tools also revolutionized the calculations and methods in

116 astronomy. 10 A senes of writings came in with Aryabhatta II Aryapak~a school,

Latadeva, the student of Aryabhatta I and author of revised (Siiryasiddhanta Siiryapa~a

school), Brahmagupta (Brahmapak~a school), Bhaskara I (AD 628), student of Aryabhatta School and a host of other scholars namely Lalla (c. AD 749), Vate8vara (AD 904), AryabhaUa II (AD 950), Sripati (AD 1039), Bhaskara II (AD 1150) and ' . . -:-· others. Aryabhatta I, an asmakfya (Keralian) lived in Magadha (modern Bihar) and wrote his Aryabha(.ta. Magadha in ancient time was a great centre of learning and is I well-known for the famous university at Nalanda (situated in the modern district of - I Patna). There was a special provision for study of astronomy in this university.

Aryabhat~a I is referred to as kulapa (= kulapati or Head of a University) by the i commentator.

Arya~iddhanta and Aryabhatyya of Aryabhatai (br ad 496) :

The Aryasiddhanta of Aryabhatta is only ·known from the quotations of Varahmihira (AD 505), Bhaskara 1 (AD 600) and Brahmagupta (AD 628) in which the day begins at midnight at Lanka. The Aryabha(.ta begins the day with sun-rise on Sunday caitra­ k~~IJiidi, Saka 421 (AD 499). A summary of contents of Aryabha(.ta will give an idea· how the kilowledge exploded. Under arithmetic, it discusses alphabetic system of notation and place-value including fundamental operations like squaring,' square root, cubing and cube root of numbers. The geometrical problems deal with area of triangle, · circle, trapezium, plane figures, volumes of right pyramid, sphere, properties of similar triangle, inscribed triangles and rectangles. Theorem of square on the diagonal, application of the properties of similar triangles. The algebra has concentrated in finding the sum of natural numbers (series method), square of n-natural numbers, cubes of n-natural numbers, formation of equation, use of rule of three for application (both direct and inverse rule), solution of quadratic equation, solution of indeterminate equation [by= ax + c, X= (by- c) I a] where solution of x and y were obtained by repeated division (ku{(aka ku( meansto pulverize)etc. In trigonometry,jya (R Sine) is

defined, and 28 jya table at an intervai of 3° 45' (R = 3438') was constructed, the value

117 ofn = 3.1416 was found to be the correct to 34 places of decimals. Aryabhatta I's value of p = 62832 I 20000 = 3 + 117 +- 1/17 + 1/11. Successive convergents are 3, 2217,

355/113, 3927/1250 which were used by la~er astronomers. In astronomy, three important hypothesis were made viz.,

(1) The mean planets revolve in geocentric circular orbits, (2) The true planets move in epicycles or in eccentric, (3) All planets have equal linear motion in their respective orbits.

The knowledge of indeteiininate equations played a significant role. The method of indeterminate equation was a successive method of division. The same method is possibly used for value of, solution of first degree and second-degree indeterminate equation. It was also used to determine the mean longitude of plane for mean longitude

= (R x A)/C, where R = revolution number of planets, A = ahargaYJa = no. of days since· the epoch and C = no. of days in a yuga or kalpa; Large number of astronomical problems ofBhaskara I are changed to (ax- c)/b = y where_ x = a~_argal']a any y =Sun's mean longitude.

Astronomical corrections and astronomical irtstruments :

The geocentric longitude of a planet is derived by the mean longitude by the following corrections.

(1) Correction for local longitude (desiintara correction). (2) Equation of the centre (biihuphalC!)· (3) Correction ofthe equation oftime due to eccentricity ofthe ecliptic. (4) Correction oflocallatitude (cara) in case of sun and moon, and an additional correction (sighraphala) in case of other planets.

Besides these, Vatesvara (904) gave lunar correction, which gives deficit of the moon's equation of centre and evection. Bhaskara II ( 1150) gave another correction, variation.

118 Mafijula (932) used a process of differentiatio~ in finding the v~locity of planet. All

siddhantic astronomy gave method and time pf eclipse, along with tithi, na~atra, kiiraf)a, yuga, since these had an importimt bearihg on religious observations.

A large number of astronomical instruments were referred to and used. To cite a few from Lalla's Si~yadhivrddhida 11 (eighth - elehnth centuries), these .are (1) air and water instrument, (2) golayantra, (3) man with a rosary of beads, (4) self-rotating wheel, and self-rotating spheres, (cakra yantra circle), (6) dhanur yantra (semi-circle), (7) kartari yantra, the scissors, (8) kapiila yantr'a (set horizontally on the ground and its ·

needle vertical), (9) bhagaf)a yant~a, (10) gha(l yantra and conversion of observed gha{fs into time only, (11) sanku yantra, (12) saliikii yantra, needle, (13) saka{a yantra

(for tithi observation), (14). ya~(hl yantra and graduated tube (for altitude, zenith distance and biihu) and others.

The extension of knowledge by Kerala astronomers :

. The knowledge of jyii (or }iva), kojyii and sara for a planet in a circle of known radius (trijyii) was used. The scholars used gradually improved value of (trijyii) where Sinus

totus = 24th jyii = R = 3438' (Aryabhatta I), 120' (Paficasiddhiinta), 3270' (Brahmagupta), 3437'44"19" (GovindasvamT, AD 850), 3437'44"48"' (Madhava c. AD

1400). From the relation C = 2nR, where C = circumference and R = radius of the circle, R was calculated. The.value ofC was taken as C = 360 degrees= 21600 minutes

and 1t = 3.1416 (.Aryabhatta I). Madhava (c. AD 1400) used a value ofn correct to II places of decimals. rJadhava used knowledge ofseries to approximate the value ofjfwi . , . . . I 3 2 2 = s = s I 3lr , and sara= s /2lr for an arcs of radius rand applied th~m repeatedly.

I Important trigonometrical relations were also found by scholars from Aryabhatta I onwards. The. successive approximation 12 of Madhava ~d other Kerala astronomers lead, for s = rx, to the discovery of

119 . 3 5 . Sm x:::;:: x- x /3! + x /5! -- .. . 2 4 Cos x = 1 - x /2! + x /4! --.. .

These were investigated and rediscovered later in Europe.

For the study of history it is important to find out the date or at least a period of any incident in the history of any country. This aspect of fixing the dates of Indian history came to importance when European scholars began their studies of Indian history. The

European scholars had no concept ~f measurement of time with the help of astronomy. They used to count the days and years from a particular king or event. Such records are not present in the ancient history of India. Therefore nobody was -~ble to fix the dates of the ancient Indian history.

It made clea~ that the Indian authors have given details of time by the astronomical records and by proper studies we can easily fix the dates of various incidents in the ancient Indian history accurately.

We think that the modem scientific calendar is very good because it is easy but it is not ·good. If a calendar from our wall is lost we are unable to find the date and month. If we happen to go to some isolated island after wrecking our ship then in four to five days we will forget the date and month and then will never be recalled. But a man like me who has studied the Indian calendar can easily tell Tithi, Nak$atra, Pak$a, Lunar months by looking at the night sky. By looking at the rising. sun f?.tu and Ayana can be told. Counting of the days can be done by Tithi and Nak.yatra. Tit hi depends on the size and shape of the Moon. Nak$atra is the star near which the Moon is seen. Every day the Moon changes its Nak$atra and completes a round in 28 days. Thus the Indian calendar . is eternal.

120 Nalcyatra tells the position of the Moon among1 the stars. Tit hi tells the distance of the Moon from the Sun, because in one Tithi the Moon goes 12 degrees away from the Sun. Lunar month is named after the Nalcyatra near which the full-moon resides. It is also the Nak$atra which rises at the Sunset and remains! visible throughout the night. From the ·name of the Lunar month the Moon's position is fixed and from it the position of the Sun can be easily calculated. The sun is exactly opposite at the full-moon while both are together at Amlivasyli.

J?_tus are six in number, three in each Ayana. Ayanas are two, Uttarliyana and Dalcyinliyana. Uttarliyana means the_ Sun's northern journey. It begins with Shis.hira f.?..tu at the Winter Solstice on 22"d December of the modern calendar. Dakshinliyana is the southern journey of the Sun which begins with the Var$li J?_tu at the Summar Solstice on ·22"d June. Var$li, Sarad, Hemanta are.the three R,tus of Dakshiriliyana, while Shishird, Vasanta and Greeshma are the three J?_tus of Uttarliyana. Each f.?..tu consists of two solar' months. Nabha and Nabhasya are the two solar months-of Var$li J?_tus, which correspond with 22"d June to 21st July and 22"d July to 21 51 August. l$li and Urjli are the two months of Sharad. !sa corresponds to 22"d August to 21 51 September. A period from 23rd September to 22 11 d October means Urja. 23rd September of the beginning of Urja is the Autumnal Equinox having equal day and night. Hemanta rtu consists of Saha (22 October to 21st November) and Sahasya (22"d . November to 21st December). 22"d December is the Winter Solstice when Tapa. the first month of J?_tu SiSira begins. The

51 second month of Sisira i.e. Tapasyli extends 1from 22"d January to 21 February. Mlidhu, the first month of Vasanta J?_tu begins on 22"d February and ends with the. Vernal Equinox on 21st March. Then Miidhava, the ·second month of Vasanta begins which ends on 21st April. From 22"d April Greeshma begins, the first month of which is $ukra. Its second month Suchi begins on 22"d May and ends on 21st June. 22"d June is

the Summer Solstice when Nabha and Var~a begin. Thus the solar seasonal months of J?_tus correspond with the modern seasonal months (Taittirzya Samhitli 4.4.1. Also Vi~hvu PuriiJJ,a Ansha 2, Adhyiiya 8, Sloka 83 ). .

121 Vi$1JU Purarza 2.8. 70 clearly speak about two solar months forming J?.tu.

Ancient authors are particular in give Tithi, Nalcyatra, lunar month and J?.tu. Sages are particular to mention the Nalcyatra of the Sun at the Winter Solatice or the Summer Solstice or the equinoxes. From these records it is evident that the ancient sages knew of the precession of Equinoxes due to which the Sun recedes in the Nalcyatra Cakra. If we know the rate of the precession of equinoxes as one degree in 72 years or one Nalcyatra . : ~· . in 960 years or one Rashi i.e. 30 degrees in 2160 years, we can calculate the year or the period of any incident about which these records are available. We get the data for I calculations if the J?.tu and lunar m

· Pancanga means five important points regarding the time. These are Tit hi, Nalcyatra, Masa, J?.tu and Ayana. In addition to these five, :the sages were particular to record the positions of the planets, which pinpoint the time.

In the epic Mahabharata, Maharshi Vyasa gives the positions of all the planets m Bhishma Paiva, Adhyaya 2,3, thus- Saturn in Puva, Jupiter in Sarvarza, Rahu in Uttara A$8.(iha and Mars in Anuradha. All these planets have a fixed rotational period. Saturn completes one rotation in 29.4545832 years. Jupiter takes 11.863013 years for one rotation. Rahu takes 18.5992 years, Mars takes 1.88089 years per rotation. Taking the respective positions of each of these planets in any year we can calculate backwards to find out when the planets were in the said positions. I found the year to be 5561 years before Christ. When Bhishma died it was Winter Solstice and he had fought for ten days and had lied on the arrow bed for 58 days. 68 days earlier than 22nd December is 16th OctQber when the Great War must have commenced. Thus we could fix the date of the Mahabharata war exactly ~s 16th October 5561 B.C. Three more planets are found described by Vyasa in the Mahabharata under the names of Shveta, Shyama and Teevra. These are Uranus, Neptune and Pluto. Calculating back we can find that the

122 respective positions get confirmed on the above date. In addition Vyasa describes two consecutive Amiivasylis, one ~aya Pa~a, two eclipses lunar and solar in one month's time and a big comet. All these evidences show the same date 161h October 5561 B.C. Eighteen mathematical points converge on this date and many other evidences point to the same date. Therefore we have to accept it as the date of the beginning of the Mahiibhiirata war.

Accepting this date we can fix the dates of about sixty incidents from the Mahiibhiirata ·including the date of exile 4th September 5574 RC. and exposure of Arjuna on 16th April5561 B.C.

Except Astronomy no other method can give such details and exact dates. In the case of the Mahiibhiirata we have to solve some riddles put forth by the sage Vyasa deliberately, but in another books no such problem arises.

IJ.tus and lunar months as described in the Mahtibhtirata point to the same time of 5561 B.C. The planet Saturn had been in the arms of Rohini according to the Mahiibhiirata. According to the calculations of late Shti Dixit S.B. this phenomenon took place for many centuries before 5294 B.C.

At the time of Raffia's birth five planets were exalted according to Viilmzki Ramliyarza. The exalted positions well-known in astrology are thus : Sun 10° in Me$ a, Mars 28° in Makara, Jupiter 5° in Karka, Venus 27° in Meena, Saturn 20° in Tulii. For calculations of thousands of years only the slow moving planets are useful and only by two planets Saturn and Jupiter we cannot fix the exact date. At least three planets are essential. In the Vtilmfki Rtimtiyarza, it is stated at Ayodhya Ktj1J¢a 4/18 that the Sun, Mars and Rahu were aspecting Dasaratha 's Nalcyatra at the time of Rama's coronation on Caitra 1 Suddha 9 \ when Rama completed 17 years of his life. In the Caitra Miisa Sun resides

in Me~a Rasi; so Rahu also must have been Me$a or just opposite in Tulii Rasi. Rahu completes one rotation in 18.5992 years, hence 17 years earlier Rahu must have been in

123 Kanyli i.e. Virgo. Calculations on this data show the date of Rama's birth as 4th December 7323 years B.C.

Instead -of coronation Rfuna had to leave for forestI in He manta Rtu. on Caitra Suddha 9th. At this occasion Valmiki states the star positions at Ayodhya 41/10, 11. These tally - ' with the modem mathematics. Rama went to the forest life on 29th November 7306 B.C. Rfuna was married to Sita on ih April 73-7 B.C,., in Bhadrapada Suddha 3rd on Uttarli I Phlilgun'f Na!cyatra.

When Hanumana returned after burning Lanka, Valmiki describes the sky vividly, which indicates the date as POU$G Krishna 1st or13rd September 7292 B.C. The war was fought from 3rd November to 15th November 7282 B.C.

I have fixed almost 40 dates of various important events from the Rlimliym:za, showing that the astronomical records are helpful in fixing dates in history.

Vedic literature records abundant astronomical data, which can fix the periods. ~igveda 10161-13 states- "Who awakened Rubhus? The Sun replied, "The Dog, because today is the end of the year." The Dog m~ans Canis major or Mrga Nalcyatra. Rubhus means clouds. Clouds were awakened by Mrga means rains began with Mrga. At present rainy season begins with Mrga. Thus a cycle of 27 Nalcyatras must have been elapsed between the present age and the Vedic age. At the rate of 960 years per Nalcyatra 27 Nalcyatras have taken 25920 years. So the Vedic statement is 25920 years old. In other word the period ofE.gveda is 23720 years B.C.

Taittirfya Samhitli 7-4-8 states that -.P.,~rva Phlilgunf is the last night of the year while Uttara Phlilgunf is the. first night of the new year, and Vasanta is the mouth of the year. This shows the Vernal equinox in the month of Phlilguna at the full Moon. Naturally, the Sun was diagonally opposite in the Bhadrapada Nalcyatra at the vernal equinox 1 showing a period of23720 years B.C. ; .

124 ~gveda 4-57-5 requests Shunasirau to shower the water made in heavens on the Earth. Shuna means god. Sirau means two heads. Two heads of dogs means Canis major and minorine Mrga Nakyatra. This suggests beginning of the rainy season at Mrga Nakyatra the period being 23720 years B.C.

The equinoxes recede backwards through Na/cyatra. So from Mrga it must have gone to Krttika. To show it there is an evidence in the names of the Stars Krttikti. The constellation Krttikti is formed of seven small stars, which are named as Ambti, Dulti,

Nitatni, Abhrayanti, Meghayani, Var~tiyanti, Cuputikti (Taittirfya Brtihma!Ja 3-1-4). All these names indicate water. AhhF~yanti mearis bringing Abhra i.e., white clouds. Meghayanti means bringing water~Joaded black clouds. Var~ayanti means bringing showers of rains. At the beginning of rainy season first white clouds of Abhra come, then Megha or water-loaded clouds come and ~hen showers come. Thus these names definitely point to the rainy season. Krttikti was bringing the Var~a /?.tu during 21800· years B.C.

Between Mrga and Krttika there is Rohi!Ji. The rainy season must have begun on Rohi!Ji some time. It is reported in the Mahtibhtirata Vana Parva 230 wherein it is also stated that the fal~ of Vega or Abhijit began because of Krttikti went to water i.e. Summer Solstice when Var$ti /?.tu began. This was the condition around 21800 years B.C. Before this, during 22760 years B.C. the Summer solstice was at the Rohi!Ji.

Apabhara!Ji is a Vedic name of Bhara!Ji Nalcyatra. It was called as Apabhara!Ji because it was filling water. How a Nalcyatra from the sky fill water on the earth? Of course by inducing rains. Thus this name was given around 20840 B.C.

The deity of Mula Nalcyatra is mentioned as Niruti. Nairutya means southwest. The sages discovered that the rainy season in India comes from southwest winds. Therefore they fixed the deity when MUla was at the summer solstice and introduced rains during 11240 B.C.

125 - ' Before Mula, Purva A\\·[u;tha used to induce rains so that its deity is fined as water during a period of 12200 B.C.

· B.gveda 1-164-19 states, "0 Jndra and Soma, you two rotating like a yoked horse are supporting this work." These rotating deities are nothing else but the solstices. That

time Jye~tha was at the summer solstice whose deity is Indra, and Mrga whose deity is Soma was at the winter solstice. This period is 10280 B.C.

Taittirfya Brahmal'}a 1-5-1-6.7 states that Krttikli to Visakhii is the northern course of the Sun while Anuradhii to Apabharal'}i was the southern course of the Sun. This clearly shows that the winter solstice was between Krttikli and Bharal'}f i.e. 26" 40' while the summer solstice was between Visakha and Anuradha at 213" 20'. On 22"d December 1995 the Sun is at 245" 50' 45" in MUla Na/cyatra on the winter solstice. It has come here from 26" 40'. So total shift back is 140" 49' 15". The rate of the precession of , equinoxes is 50.2 per year. So it is seen that 10098. 704 years· ago the statement is written.1t comes to 8103 years B.C.

Asvalayana Gryhya Sutra 2-3-5 advises to meditate on Hemanta f?.tu on Margasfr$a Pun:zima. Naturally, Hemanta was ending at Magha Pun;ima on the winter solstice. The full Moon at Magha shows the Sun to be at 307" at winter solistice. At present the winter solstice occurs at 246". It shows that the Sun has receded back by 61". The

precession rate is 72 years per degree. So 61X 72 = 4392 years ago Asvaliiyana Gryhya sutra is written. It shows 2404 B.C. at least 1900 B.C. as its period.

SusrutaSamhita 6-10 tells rainy season in Bhadrhpad~ and Asvina Masas. It was the condition around 400 B.C. The Samhita also states that Magha Masa and Sishira Stu began together. This indicates 2000 B.C. Thus it appears that the first edition of SusrutaSamhita was formed in 4000 B.C. while the second edition was prepared at 2000 B.C.

Maitraym;i Upani$ad 6/14 records the summer solstice at the beginning of the Magha

Nalcyatra or at ·120°. At present it is at 65° 40' 42~'. The difference is of 54 ° 19' 18" or

126 195558 seconds. The precession rate is 50.42" per year. It shows that MaitriiyalJi

Upani~ad is composed 3895,5776 years ago or 1909 B.C.

Vishnu Purfu).a tells at 2/8/66 to 76 that equal day and night come when the Sun entered

Me~a or Tula Riisi. It shows a period not later than 1609 B.C.

Vedlinga Jyoti~a shows the winter solstice on Dhani.ythii which cannot be later that 1640 B.C.

Parasara tells Summer solstice at. middle of Asle~a. at 113 ° 20'. Its date comes to 1159 B.C.

Kausitakf or Sankhliylina BrahmalJa 113 tells that in the middle of the rainy season one should see at the Punarvasu Nalcyatra while offering oblation. But in these days in the first fortnight the Moon does not combine with Punarvasu. Therefore give oblation on ' · Amlivasya, which comes after A~a4ha, because ~n the. Amavasya there is Purnavasu Nalcyatra.

In the first fortnight of Punarvasu is invisible in the months of Pau$a to A$a4ha. These

are omitted. ~t shows that Jeshtha and A~acjha were not the months of rainy season It is

suggested to useAmavasyii coming after A~iicjha. Thus it appears that the rainy season used to begin from Srtivarza Mlisa. At present the rainy season begins from Je.y{ha Masa. The rainy season has shifted back by two months. It means that the Sun has receded 60°. At the rate of 72 years per degree the shift must have taken place in 4320 years. Thus Kausitaki Briihmana appears to be written around 2320 B.C.

Matsya Purava 20415 tells that if one oblates on the Trayodasi with Magha Nalcyatra during Var$li .etu he gets benefited. Magha Nak.yatra on Trayodasi comes in Siirvarza on Bhadrapada mlisa. Thus these two were the months of Var$G f?.tu when Miitsya Pura7Ja is composed. The period is around 200 B.C:.

127 Shrimad Bhagavata Purcil;a shows equal day and night when the Sun was in Me.~a or Tufa Rasi. It shows the time to be 1600 to 2000 B.C.

Kalidasa describes Var$CI I?.tu from A$Ciqha first day. From that he appears to be at the beginning of the Christian era i.e. 2000 years ago since today.

Varahmihira gives the summer solstice at Karkadya i.e. at 90°. So his date is 520 A.D.

Thus all the steps of one to two thousand years are shown by Astronomy. This cannot be done by Archaeology or linguistics of any other science or method. There is a gap

from 12000 B.C. to 20000 B.C. wh~ch can be filled up by stray evidences such as the fall of Vega (Abhijit) mentioned in the Mahabharata which took place around 13000 B.C. Valmrki Ramaya.I).a states that Daityakula had Mula as their Nak.}atra and Jkyvaku Kula Na/cyatra was Visakha. It means that Daitya Kula began when Vernal equinox was at MUla i.e. 17000 B.C. Jlcyvaku Kula started with Visakha at the Vernal equinox around . 15000 B.C. Thus all the steps. of history of the ancient Indian culture are seen ranging . from 25000 RC. to 520 A.D. Only Astronomy can give such fineries about time.

128 NOTES AND REFERENCES

. I

1. Vediinga Jyoti~a, edited and translated by T.S. Kuppanna Sastry ·along with RK and Yiijus. Recensions, text I and 2, Indian National Science Academy, 1985, pp. 35-36.

2. B. Datta, The Science of Sulba, Calcutta University, I 932 (reprinted); S.N. Sen, and A.K. Bag, 'The Sulbasiitras of Baudhiiyana, A.pastamba, Katyayana and Manava', with Eng. translation and commentary, Indian National S_cience Academy, New Delhi, 1983.

3. A.K. Bag, 'Ritual Geometry in India and its Parallelism in Other Cultural Areas', Indian Journal ofHistory ofScience, 25 ( 1-4 ), 1990, pp. 4-19.

4. George Sarton, 'The Appreciation of Ancient and Medieval . Science during the Renaissance, ' 1450-1600', Philadelphia Univ., and Pennsylvania Press, J955, p.l51.

5. B. Datta, 'Vedic Mathematics', The Cu/turaiHeritage ofIndia, Vol. 3, Calcutta, 1937, pp. 378- 401; R.N. Mukherjee, 'Background to the Discovery of Zero', Indian Journal of History of Science, I2 · (2), 1977, pp. 225-3 I; A.K. Bag, Mathematics in Ancient and Medieval India,

Chau~amba Orientalia, Varanasi, I 979.

6. B.A. Rosenfeld, 'AI Khwarizmi and Indian Science' in Interaction Between Indian and Central Asian Science and Technology in Medieval Times, Vol. I, Indian National Science Academy, New Delhi, I990, p. I32.

7. A.K. Bag, Science and Civilization in India (Harappan period), Vol. I, Navrang, New Delhi, I985, pp. I02-07.

8. S.B. Dikshit, Bhiiratlya Jyotish Siistra (in Marathi), Poomi, 1896, translated into Eng. by R.V. Vaidya, Pt. I (Vedic and Vediinga Period), Delhi,• 1969, Pt. II, Siddhiintic and Modern Periods, I 1981.

9. K.S. Shukla, 'The Yuga ·of the Yavana-Jataka - David Pingree's Text and Translation Reviewed', Indian Journal ofHistory ofScience, 24(4), 1989, pp. 211-23. I

129 10. K.S. Shukla, 'The Paficaiddhiintikii of Variihamihira (I)' Indian ,Journal of History of Science, 9(1), 1974, pp. 62-76; T.S. Kuppanna Sastry, Paiicasiddhiintikii of Variihamihira, with translation and notes, P.P.S.T. Foundation, Advar, Madras, 1992.

II. Bina Chatterjee·, Sisyadhiw:ddhida Tantra of Lalla, edited and translated into Eng., 2 vols., Indian National Science Academy, New Delhi, 1981; sec Vol. 2, pp. 279-91.

12. T.A. Saraswati, 'Development of Mathematical Ideas in India', Indian Journal of History of Science, Nos. I and 2, 1969, pp. 59-78.

130